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Let $\mathbf{D}_{3}$ be a bigraded 3-decorated disk with an arc system$\mathbf{A}$. We associate a bigraded simple closed arc$\widehat{\eta}_{\frac{r}{s}}$ on $\mathbf{D}_{3}$ to any rational number$\frac{r}{s}\in\overline{\mathbb{Q}}=\mathbb{Q}\cup\{\infty\}$. We show thatthe right (resp. left) $q$-deformed rational numbers associated to$\frac{r}{s}$, in the sense of Morier-Genoud-Ovsienko (resp.Bapat-Becker-Licata) can be naturally calculated by the$\mathfrak{q}$-intersection between $\widehat{\eta}_{\frac{r}{s}}$ and$\mathbf{A}$ (resp. dual arc system $\mathbf{A}^*$). The Jones polynomials ofrational knots can be also given by such intersections. Moreover, thecategorification of $\widehat{\eta}_{\frac{r}{s}}$ is given by the sphericalobject $X_{\frac{r}{s}}$ in the Calabi-Yau-$\mathbb{X}$ category of Ginzburgdga of type $A_2$. Reduce to CY-2 case, we recover result ofBapat-Becker-Licata with a slight improvement.

Topological model for q-deformed rational number and categorification